Tipping Points in Macroeconomic Agent-Based Models
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Tipping points in macroeconomic Agent-Based models Stanislao Gualdi,1 Marco Tarzia,2 Francesco Zamponi,3 and Jean-Philippe Bouchaud4 1 Universit´ePierre et Marie Curie - Paris 6, Laboratoire de Physique Th´eoriquede la Mati`ere Condens´ee, 4, Place Jussieu, Tour 12, 75252 Paris Cedex 05, France and IRAMIS, CEA-Saclay, 91191 Gif sur Yvette Cedex, France ∗ 2 Universit´ePierre et Marie Curie - Paris 6, Laboratoire de Physique Th´eoriquede la Mati`ere Condens´ee, 4, Place Jussieu, Tour 12, 75252 Paris Cedex 05, France 3Laboratoire de Physique Th´eorique, Ecole´ Normale Sup´erieure, UMR 8549 CNRS, 24 Rue Lhomond, 75231 Paris Cedex 05, France 4CFM, 23 rue de l'Universit´e,75007 Paris, France, and Ecole Polytechnique, 91120 Palaiseau, France. The aim of this work is to explore the possible types of phenomena that simple macroeconomic Agent-Based models (ABM) can reproduce. Our motivation is to understand the large macro- economic fluctuations observed in the \Mark I" ABM devised by D. Delli Gatti and collaborators. Our major finding is the existence of a first order (discontinuous) phase transition between a \good economy" where unemployment is low, and a \bad economy" where unemployment is high. We show that this transition is robust against many modifications of the model, and is induced by an asymmetry between the rate of hiring and the rate of firing of the firms. This asymmetry is induced, in Mark I, by the interest rate. As the interest rate increases, the firms become more and more reluctant to take further loans. The unemployment level remains small until a tipping point beyond which the economy suddenly collapses. If the parameters are such that the system is close to this transition, any small fluctuation is amplified as the system jumps between the two equilibria. We have also explored several natural extensions. One is to allow this hiring/firing propensity to depend on the financial fragility of firms. Quite interestingly, we find that in this case, the above transition survives but becomes second order. We also studied simple wage policies and confidence feedback effects, whereby higher unemployment increases the saving propensity of households. We observe several interesting effects, such as the appearance of acute endogenous crises, during which the unemployment rate shoots up before the economy recovers. We end the paper with general comments on the usefulness of ABMs to model macroeconomic phenomena, in particular in view of the time needed to reach a steady state. It is human nature to think wisely and to act absurdly { Anatole France I. INTRODUCTION: FROM MICRO-RULES TO MACRO-BEHAVIOUR Inferring the behaviour of large assemblies from the behaviour of its elementary constituents is arguably one of the most important problems in a host of different disciplines: physics, material sciences, biology, computer sciences, sociology and, of course, economics and finance. It is also a notoriously hard problem; statistical physics has developed in last 150 years essentially to understand this micro-macro link. Clearly, when interactions are absent or small enough, the system as a whole merely reflects the properties of individual entities. This is the canvas of traditional macro- economic approaches. Economic agents are assumed to be identical, non-interacting, rational agents, an idealization known as the \Representative Agent" (RA). In this framework, micro and macro trivially coincide. However, we know (in particular from physics) that discreteness, heterogeneities and/or interactions can lead to totally unexpected phenomena. Think for example of super-conductivity or super-fluidity: before their experimental discovery, it was simply beyond human imagination that individual electrons or atoms could \conspire" to create a collective state that can flow without friction. Micro and macro behaviour not only do not coincide in general, but genuinely surprising behaviour can emerge through aggregation. From the point of view of economic theory, this is interesting, because financial and economic history is strewn with bubbles, crashes, crises and upheavals of all sorts. These are very hard to fathom within a Representative Agent framework [1], within which crises would require large aggregate shocks, when in fact small local shocks can trigger large systemic effects when heterogeneities, interactions and network effects are taken into account [2, 3]. The need to include these effects has spurred a large activity in \Agent-Based models" ∗Electronic address: [email protected] 2 (ABMs) [4{6]. These models need numerical simulations, but are extremely versatile because any possible behavioural rules, interactions, heterogeneities can be taken into account. In fact, these models are so versatile that they suffer from the \wilderness of high dimensional spaces" (paraphrasing Sims [7]). The number of parameters and explicit or implicit choices of behavioural rules is so large (∼ 10 or more, even in the simplest models, see below) that the results of the model may appear unreliable and arbitrary, and the calibration of the parameters is an hopeless (or highly unstable) task. Mainstream RA models, on the other hand, are simple enough to lead to closed form analytical results, with simple narratives and well trodden calibration avenues. In spite of their unrealistic character, these traditional models are still favoured by most economists, both in academia and in institutional and professional circles. ABMs are seen at best as a promising research direction and at worst as an unwarranted \black box" (see [8] for an enlightening discussion on the debate between traditional DSGE models and ABMs, and [9, 10] for further insights). At this stage, it seems to us that some clarifications are indeed needed, concerning both the objectives and method- ology of Agent-Based models. ABMs do indeed suffer from the wilderness of high dimensional spaces, and some guidance is necessary to put these models on a firm footing. In this respect, statistical physics offers a key concept: the phase diagram in parameter space [11]. A classic example, shown in Fig. 1, is the phase diagram of usual substances as a function of two parameters, here temperature and pressure. The generic picture is that the number of distinct phases is usually small (e.g. three in the example of Fig. 1: solid, liquid, gas). Well within each phase, the properties are qualitatively similar and small changes of parameters have a small effect. Macroscopic (aggregate) properties do not fluctuate any more for very large systems and are robust against changes of microscopic details. This is the \nice" scenario, where the dynamics of the system can be described in terms of a small number of macroscopic (aggregate) variables, with some “effective" parameters that encode the microscopic details. Other scenarios are of course possible; for example, if one sits close to the boundary between two phases, fluctuations can remain large even for large systems and small changes of parameters can radically change the macroscopic behaviour of the system. There may be mechanisms naturally driving the system close to criticality (like in Self Organized Criticality [12]), or, alternatively, situations in which whole phases are critical, like for \spin-glasses" [13]. In any case, the very first step in exploring the properties of an Agent-Based model should be, we believe, to identify the different possible phases in parameter space and the location of the phase boundaries. In order to do this, numerical simulations turn out to be very helpful [14, 15]: aggregate behaviour usually quickly sets in, even for small sizes. Some parameters usually turn out to be crucial, while others are found to play little role. This is useful to establish a qualitative phenomenology of the model { what kind of behaviour can the model reproduce, which basic mechanisms are important, which effects are potentially missing. This first, qualitative step allows one to unveil the \skeleton" of the ABM. Simplified models that retain most of the phenomenology can then be constructed and perhaps solved analytically, enhancing the understanding of the important mechanisms, and providing some narrative to make sense of the observed effects. In our opinion, calibration of an ABM using real data can only start to make sense after this initial qualitative investigation is in full command { which is in itself not trivial when the number of parameters is large. The phase diagram of the model allows one to restrict the region of parameters that lead to \reasonable" outcomes (see for example the discussion in [16, 17]). The aim of this paper is to put these ideas into practice in the context of a well-studied macroeconomic Agent-Based model (called \Mark I" below), devised by Delli Gatti and collaborators [18, 19]. This model is at the core of the European project \CRISIS", which justifies our attempt to shed some theoretical light on this framework.1 In the first part of the paper, we briefly recall the main ingredients of the model and show that as one increases the baseline interest rate, there is a phase transition between a \good" state of the economy, where unemployment is low and a \bad" state of the economy where production and demand collapse. In the second part of the paper, we propose a highly simplified version of Mark I that aims at capturing the main drivers of this phase transition. We show that the most important parameter is the asymmetry between the firms’ propensity to hire (when business is good) or to fire (when business is bad). In Mark I, this asymmetry is induced by the reaction of firms to the level of interest rates, but other plausible mechanisms would lead to the same effect. The simplest version of the model is amenable to an analytic treatment and exhibits a \tipping point" (i.e. a discontinuous transition) between high employment and high unemployment. Enriching the model by, e.g. endowing firms with a strategy that depends on their financial wealth, leads to a modified picture, where the level of unemployment is now continuous, and where low frequency fluctuations of the employment rate/production appear due to the proximity of a second order transition.